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A055216
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Triangle T(n,k) by rows, n >= 0, 0<=k<=n: T(n,k) = Sum( binomial(n-k,i)*Sum( binomial(i,j),j=0..k-i).
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7
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1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 3, 1, 1, 5, 10, 8, 3, 1, 1, 6, 15, 17, 9, 3, 1, 1, 7, 21, 31, 23, 9, 3, 1, 1, 8, 28, 51, 50, 26, 9, 3, 1, 1, 9, 36, 78, 96, 66, 27, 9, 3, 1, 1, 10, 45, 113, 168, 147, 76, 27, 9, 3, 1, 1, 11, 55, 157, 274, 294, 192, 80, 27
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| T(n,k) = maximal number of different sequences that can be obtained from a ternary sequence of length n by deleting k symbols.
T(i,j) = number of paths from (0,0) to (i-j,j) using steps (1 unit right) or (1 unit right and 1 unit up) or (1 unit right and 2 units up).
If m >= 1 and n >= 2, then T(m+n-1,m) is the number of strings (s(1),s(2),...,s(n)) of nonnegative integers satisfying s(n)=m and 0<=s(k)-s(k-1)<=2 for k=2,3,...,n.
T(n,k) = number of 1100-avoiding 0-1 sequences of length n containing k good 1's. A 1 is bad if it is immediately followed by two or more 1's and then a 0; otherwise it is good. In particular, a 1 with no 0's to its right is good. For example, 110101110111 is 1100-avoiding and only the 1 in position 6 is bad and T(4,3) counts 0111, 1011, 1101. - David Callan (callan(AT)stat.wisc.edu), Jul 25 2005
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REFERENCES
| D. S. Hirschberg, Algorithms for the longest subsequence problem, J. ACM, 24 (1977), 664-675.
V. I. Levenshtein, Efficient reconstruction of sequences from their subsequences or supersequences, J. Combin. Theory Ser. A 93 (2001), no. 2, 310-332.
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FORMULA
| T(i, 0)=T(i, i)=1 for i >= 0; T(i, 1)=T(i, i-1)=i for i >= 2; T(i, j)=T(i-1, j)+T(i-2, j-1)+T(i-3, j-2) for 2<=j<=i-2, i >= 3.
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EXAMPLE
| 8=T(5,2) counts these strings: 013, 023, 113, 123, 133, 223, 233, 333.
Rows: {1}; {1,1}; {1,2,1}; {1,3,3,1}; {1,4,6,3,1} ...
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CROSSREFS
| Row sums: A008937. Central numbers: T(2n, n)=A027914(n) for n >= 0.
Sequence in context: A157283 A067049 A090641 * A180181 A128629 A107065
Adjacent sequences: A055213 A055214 A055215 * A055217 A055218 A055219
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KEYWORD
| nonn,tabl
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu), May 07 2000
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EXTENSIONS
| Better description and references from N. J. A. Sloane (njas(AT)research.att.com), Aug 05 2000
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